Pre-Calc Homework Solutions 330

# Pre-Calc Homework Solutions 330 - 330 Section 8.2 ex ex 2xe...

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9. (a) f ( x ) 5 } e x 1 e x x 2 } 5 1 1 } x e 2 x } f 9 ( x ) 5 } 2 xe x e 2 2 x x 2 e x } 5 } 2 x e 2 x x 2 } } 2 x e 2 x x 2 } 5 0 x (2 2 x ) 5 0 x 5 0 or x 5 2 f 9 ( x ) , 0 for x , 0 or x . 2 The graph decreases, increases, and then decreases. f (0) 5 1; f (2) 5 1 1 } e 4 2 } < 1.541 f has a local maximum at < (2, 1.541) and has a local minimum at (0, 1). (b) f is increasing on [0, 2] (c) f is decreasing on ( 2‘ , 0] and [2, ). 10. f ( x ) 5 } x 1 x sin x } 5 1 1 } sin x x } , x 0 Observe that ) } sin x x } ) , 1 since ) sin x ) , ) x ) for x 0. lim x 0 f ( x ) 5 1 1 lim x 0 } sin x x } 5 1 1 1 5 2 Thus the values of f get close to 2 as x gets close to 0, so f doesn’t have an absolute maximum value. f is not defined at 0. Section 8.2 Exercises 1. lim x } x 3 2 e 3 x x 1 1 } 5 lim x } 3 x 2 e 2 x 3 } 5 lim x } 6 e x x } 5 lim x } e 6 x } 5 0 x 3 2 3 x 1 1 grows slower than e x as x . 2. First observe that ˇ 1 w 1 w x w 4 w grows at the same rate as x 2 . lim x } ˇ 1 w x 1 w 2 x w 4 w } 5 lim x !
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